The Geometry of Rectangular Multisets

TL;DR

This paper introduces a piecewise Euclidean bi-simplicial cell structure for the space of multisets within a rectangle, linking it to complex polynomials and permutahedra.

Contribution

It presents a novel geometric cell structure for multiset spaces and explores their connections to well-known mathematical objects.

Findings

Established a bi-simplicial cell structure for multiset spaces

Connected multiset spaces to complex polynomials and permutahedra

Provided a geometric framework for analyzing multiset configurations

Abstract

This article describes a natural piecewise Euclidean bi-simplicial cell structure for the space of $n$-element multisets in a fixed Euclidean rectangle. In particular, we highlight some connections with spaces of complex polynomials and permutahedra.